Algebraically closed field: Difference between revisions
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imported>1234qwer1234qwer4 closed, not complete |
imported>Filtron11 m →Other properties: added remark on cardinality |
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===Polynomials of prime degree have roots=== | ===Polynomials of prime degree have roots=== | ||
If every polynomial over ''F'' of prime degree has a root in ''F'', then every non-constant polynomial has a root in ''F''. | If every polynomial over ''F'' of prime degree has a root in ''F'', then every non-constant polynomial has a root in ''F''.{{sfn|Shipman|2007}} It follows that a field is algebraically closed if and only if every polynomial over ''F'' of prime degree has a root in ''F''. | ||
===The field has no proper algebraic extension=== | ===The field has no proper algebraic extension=== | ||
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The theory of algebraically closed fields has [[quantifier elimination]]. | The theory of algebraically closed fields has [[quantifier elimination]]. | ||
There is no algebraically closed [[finite field]]: if there were such a field, with underlying set <math>\{0, 1, k_2, ..., k_n\}</math> for some <math>n\in\N</math>, then the polynomial <math>x(x-1)(x-k_2)...(x-k_n) +1</math> would never vanish for any value of <math>x</math>. | |||
== See also == | == See also == | ||
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{{DEFAULTSORT:Algebraically Closed Field}} | {{DEFAULTSORT:Algebraically Closed Field}} | ||
[[Category:Field | [[Category:Field theory]] | ||